For simplicity, we give the proof only for \(N=2\) , but the proof in the general
case for \(N\)  is similar. We start with the inequality
\(\Gamma (F:\delta _{1}+\delta _{2},\eta _{1}+\eta _{2})_{p}\le \Gamma (F:\delta _{1},\eta _{1})_{p}+\Gamma (F:\delta _{2},\eta _{2})_{p}. \) 
Indeed, let \(t_{1},r_{1},s_{1}\)  with \(|t_{1}-s_{1}|\le \delta _{1}+\delta _{2}\) ,
\(|t_{1}-r_{1}|\le \delta _{1}\) , \(|r_{1}-s_{1}|\le \delta _{2}\)  and \(t_{2},r_{2},s_{2}\)  with \(|t_{2}-s_{2}|\le \eta _{1}+\eta _{2}\) , \(|t_{2}-r_{2}|\le \eta _{1}\) , \(|r_{2}-s_{2}|\le \eta _{2}\) .
Since \(\mu \)  is submodular, the Minkowski inequality is available in the vector
space \({\mathcal {L}}(\Omega ,{\mathcal {A}},\mu )\) ; see Theorem 2, p. 5 in
[1]}, or Proposition 9.4, p. 109-110 in [2]}. Therefore
\(\left( (C)\int _{\Omega }|F(t_{1},t_{2},\omega )-F(s_{1},s_{2},\omega )|^{p}d\mu (\omega )\right) ^{1/p}\\\le \left( (C)\int _{\Omega }|F(t_{1},t_{2},\omega )-F(r_{1},r_{2},\omega )|^{p}d\mu (\omega )\right) ^{1/p}\\+\left( (C)\int _{\Omega }|F(r_{1},r_{2},\omega )-F(s_{1},s_{2},\omega )|^{p}d\mu (\omega )\right) ^{1/p}.\) 
Passing now to the corresponding suprema, first in the right-hand side and
then in the left-hand side, we are led to (REF ). As a consequence
\(\Gamma (F:n\delta ,m\eta )_{p}\le \max \lbrace n,m\rbrace \Gamma (F:\delta ,\eta )_{p}\) 
and taking into account that \(\alpha <[\alpha ]+1,~\beta <[\beta ]+1\)  and
\(\max \lbrace [\alpha ]+1,[\beta ]+1\rbrace =\max \lbrace [\alpha ],[\beta ]\rbrace +1<\alpha +\beta +1,\) 
one easily obtain the inequality in the statement of Theorem REF .
